A (continuous, linear) operator \(T\)  acting on a separable Fréchet space \(\mathcal {X}\)  is said to be hypercyclic if there exists \(f\in \mathcal {X}\)  such that \(\lbrace T^nf\rbrace _{n\ge 1}\)  is dense in \(\mathcal {X}\) .
In this case we say that the vector \(f\)  is hypercyclic for \(T\) . To say that \(T\)  is supercyclic means that there exists a vector \(x\in \mathcal {X}\)  such that the projective orbit \(\lbrace \lambda T^nx\,\,:\,\,n\in \mathbb {N},\lambda \in \mathbb {C}\rbrace \)  is dense in \(\mathcal {X}\) .
The operator \(T\)  is said to have a hypercyclic (supercyclic) subspace if there exists an infinite dimensional closed subspace whose non-zero vectors are hypercyclic (supercyclic) for \(T\) . The study of hypercyclic (supercyclic) subspaces is a mainstream research line in the theory of hypercyclic operators (see [1]} Chapter 10, [2]} Chapter 8, [3]}), and its interest come from the invariant subspace problem.
